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Calculating div(theta) and tangent curves

  1. Sep 11, 2014 #1
    1. The problem statement, all variables and given/known data

    Calculate ∇Θ where [tex]Θ(x)=\frac{\vec{p} \cdot \vec{x}}{r^3}[/tex]. Here [tex]\vec{p}[/tex] is a constant vector and [tex]r=|\vec{x}|[/tex]. In addition, sketch the tangent curves of the vector function ∇Θ for [tex]\vec{p}=p\hat{z}[/tex]

    (b) Calculate [tex]∇ (cross) A → \vec{A}=\frac{\vec{m}x\vec{X}}{r^3}[/tex] m is constant vector. Sketch the tangent curves of ∇(cross)A for [tex]\vec{m}=m\hat{z}[/tex]

    2. Relevant equations

    gradient vector

    3. The attempt at a solution

    Well when I apply the gradient vector to the function Θ I get many terms and a very ugly answer. I am not sure if this will clean up nicely? Is there an easier way of doing this then brute force? Also, I am not sure how to represent the tangent curve of the vector function [tex]\vec{p}=p\hat{z}[/tex] or [tex]\vec{m}=m\hat{z}[/tex].
     
  2. jcsd
  3. Sep 12, 2014 #2

    vanhees71

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    2016 Award

    Sometimes the Ricci calculus is easier than the nabla calculus. BTW: Both questions are not about div but about grad and curl.

    For the first problem you have to calculate (Einstein summation convention implied)

    [tex]\partial_j \Theta=\partial_j \left ( \frac{p_k x_k}{r^3} \right ).[/tex]
    This is now just the task to take the partial derivatives using the usual rules for differentiation (product rule in this case).

    For the second problem note that
    [tex](\vec{\nabla} \times \vec{A})_j = \epsilon_{jkl} \partial_k A_l,[/tex]
    where [itex]\epsilon_{jkl}[/itex] is the Levi-Civita symbol.
     
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